Interesting question. not my area of expertise but i have a feew comments to make and hope they may be useful.
It sounds like a contact dermatitis (irritant) rather than an allergy.
Chemical maintenance of pools involves many chemicals other than chlorine (pH+ pH-, flocculants etc) but from your very clear history in your OP chlorine does sound to be the "troublemaker."
Yes, I would suggest that you seek advice from a medically qualified allergist, but be careful as many doctors just "play at it." Your GP should be you in touch with an allergy clinic, but I'm afraid there are many types of chlorine compounds but only one chlorine element, so it is either chlorine or it isn't.
It isn't easy to get to know from the pool cleaner at a hotel or villa when you go on vacation as to the levels of chlorine and unlikely to be of use in any case.
Good luck